A new method of bandwidth selection for kernel density estimators is proposed. The method, termed indirect cross-validation, or ICV, makes use of so-called selection kernels. Least squares cross-validation (LSCV) is used to select the bandwidth of a selection-kernel estimator, and this bandwidth is appropriately rescaled for use in a Gaussian kernel estimator. The proposed selection kernels are linear combinations of two Gaussian kernels, and need not be unimodal or positive. Theory is developed showing that the relative error of ICV bandwidths can converge to 0 at a rate of n −1/4 , which is substantially better than the n −1/10 rate of LSCV. Interestingly, the selection kernels that are best for purposes of bandwidth selection are very poor if used to actually estimate the density function. This property appears to be part of the larger and well-documented paradox to the effect that "the harder the estimation problem, the better cross-validation performs." The ICV method uniformly outperforms LSCV in a simulation study, a real data example, and a simulated example in which bandwidths are chosen locally.
In this paper we provide insight into the empirical properties of indirect crossvalidation (ICV), a new method of bandwidth selection for kernel density estimators.First, we describe the method and report on the theoretical results used to develop a practical-purpose model for certain ICV parameters. Next, we provide a detailed description of a numerical study which shows that the ICV method usually outperforms least squares cross-validation (LSCV) in finite samples. One of the major advantages of ICV is its increased stability compared to LSCV. Two real data examples show the benefit of using both ICV and a local version of ICV.
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